The areas of three of the faces of a right, rectangular prism are $24 \hspace{.6mm} \mathrm{cm}^2$, $32 \hspace{.6mm} \mathrm{cm}^2$, and $48 \hspace{.6mm} \mathrm{cm}^2$. What is the volume of the prism, in cubic centimeters?
Answer: Let the three dimensions of the prism (length, width, and height, although not necessarily in that order) be $x,y,z$ such that $xy = 24$, $xz = 32$, and $yz = 48$. Then the volume of the prism is $xyz$. Multiplying our three equations together and taking the square root of both sides, we find that $xyz = \sqrt{24\cdot 32\cdot 48} = \boxed{192}$ cubic centimeters.